Probabilistic Stand Still Detection using Foot Mounted IMU
|
|
- Ella Valerie Summers
- 6 years ago
- Views:
Transcription
1 Probbilistic Stnd Still Detection using Foot Mounted IMU Jons Cllmer, Dvid Törnqvist nd Fredrik Gustfsson Division of Automtic Control Linköping University Linköping, Sweden {cllmer, tornqvist, Abstrct We consider stnd still detection for indoor locliztion bsed on observtions from footmounted inertil mesurement unit (IMU). The min contribution is sttisticl frmework for stnd-still detection, which is fundmentl step in zero velocity updte (ZUPT) to reduce the drift from cubic to liner in time. First, the observtions re trnsformed to test sttistic hving non-centrl chi-squre distribution during zero velocity. Second, hidden Mrkov model is used to describe the mode switching between stnd still, wlking, running, crwling nd other possible movements. The resulting lgorithm computes the probbility of being in ech mode, nd it is esily extendble to dynmic nvigtion frmework where mp informtion cn be included. Results of first mode probbility estimtion, second mp mtching without ZUPT nd third step length estimtion with ZUPT re provided. Keywords: Indoor locliztion, stnd still detection, HMM, ZUPT Introduction The problem of indoor locliztion hs received n incresing mount of ttention in the lst couple of yers [, 2, 3, 5, 4, 8]. The desire to ccurtely trck the position of first responders or militry personel, or to provide positioning id for civilins in shopping mlls nd irports, hs led to trnsition from robot sensor pltforms to humn ones. To trck person, vriety of sensors cn be used. A foot or body mounted IMU with ccelerometers, gyros nd mgnetometers is simple nd chep nd is therefore very common sensor. It is usully supported by rnge mesuring rdio device such s WiFi or UWB [8] or is fused with preexisting mps for enhnced trcking precision [8]. The IMUs used for indoor locliztion re smll nd chep nd consequently perform quite poorly. There is commonly drift in the gyros cusing the orienttion est west t = south north Figure : Locliztion experiment without ZUPT. Red dots re the position hypotheses nd the blue dot nd line is the men position nd the heding mesurement, respectively. The lck of ZUPT mens tht we hve lrge step length uncertinty, which cuses the hypotheses to spred ll long the corridor. estimte to be incorrect. Since the orienttion is wrong, the direction of down is wrong nd prt of the grvittionl ccelertion will insted be believed to originte from the user moving the sensor. This error is double integrted to estimte the sensor position, resulting in position error tht grows cubiclly in time. This rpidly cuses very lrge positioning errors. The gyro drift in foot mounted IMU cn be corrected if we cn detect tht the foot is on the ground. Then the foot is sttionry nd the gyro should be showing zero ngulr velocity. Insted it is showing the drift which cn be estimted nd then compensted for. This is known s ZUPT nd reduces the positioning error to being liner in time [3]. Previously, stnd still detection hs been performed d hoc, usully by compring the signl to threshold. In this work we put the stnd still detection in probbilistic frmework using test sttistics with known dis-
2 tributions nd Hidden Mrkov Model (HMM). The result is probbility of stnd still t every time instnt which cn be used for ZUPT in filtering frmework. Figure shows n illustrtive exmple of mp ided locliztion experiment without ZUPT. No stnd still detections results in very uncertin step length estimtes cusing the position hypotheses to spred ll long the corridor. 2 Relted Work Most solutions to the stnd still detection problem use n verged ccelerometer or gyro vlue nd compre it to threshold [, 2, 3, 5]. The threshold is chosen d hoc nd is normlly quite low to minimize flse positives. Another pproch is the moving vrince used in [4] where the vrince computed over sliding window is compred to threshold. Probbilistic zero velocity detection hs previously been proposed in [6] who used hypothesis test to determine if the foot ws sttionry or moving. The hypothesis test ws performed using test sttistic bsed on Generlized Likelihood Rtio Test (GLRT). The pdf of the ccelertion nd/or the ngulr velocity during the swing phse of the step, ws pproximted with n unnormlized uniform distribution. The pdf during stnd still ws bsed on the exponentil of the norm of the ccelertion nd/or the ngulr velocity, which hs n unknown distribution. The resulting test sttistic ws moving verge of the norm of the ccelertion mesurements nd/or the ngulr velocity mesurements. This ws compred to threshold to determine if the foot ws to be rendered sttionry. Since the test sttistic hs n unknown distribution the threshold ws chosen d hoc, mking the frmework similr to the ones in [, 2, 3, 5]. The test sttistics used in [6] re similr to the ones used in this work since we both evlute three different ones where one is ccelertion bsed, one is ngulr velocity bsed nd one is bsed on combintion of ccelertion nd ngulr velocity. The ccelertion bsed test sttistic differs though in tht we hve chosen one which hs known distribution. This is lso the cse in the combined test sttistic. Our frmework to determine the mode probbilities lso differs. 3 Stnd Still Detection The sensor is n Xsens MT motion sensor smpling t Hz. The signls used re the ccelerometers nd the gyros. An exmple of wlking sequence is shown in Figure 2. The foot is sttionry in the time instnts round 55, 66, 77, 87 nd 98. During these phses the norm of the ccelerometer signls is the grvittion constnt with noise. Simultneously, the norm of the ngulr velocity signl is zero with some dditive noise. 3 2 Accelerometer dt Gyro dt Figure 2: Exmple of ccelerometer dt (where x, y nd z is blue, green nd red) nd gyro dt (ω x, ω y nd ω z is blue, green nd red, ω i is ngulr rottion rte round xis i) during wlking sequence. The foot is sttionry round time instnts 55, 66, 77, 87 nd Sensor Models The signl model is y t = [ ] [ ] y t (θ) v y ω + t t (θ) v ω t () where y t nd y ω t denote ccelertion vector nd ngulr velocity vector, respectively. Further, θ denotes the model dependence of the phse of the humn step sequence. Nturlly, the model differs significntly between when the foot is t stnd still nd when it is swinging. The mesurements hve dditive Gussin noise distributions v N(, σ 2 ) nd v ω N(, σ 2 ω). The noise covrinces re independent, resulting in σ 2 ω = σωi 2 nd σ 2 = σi, 2 where I is the 3x3 identity mtrix. During stnd still the sensor model cn be described s [ ] [ ] [ ] y t gu v y ω = + t t v ω, (2) t where u is grvittionl direction vector nd g is the grvittionl constnt 9.8. When the foot is moving the sensor model chnges to [ ] y t = y ω t [ ] [ ] gu + t v + t ω t v ω t (3) where t nd ω t hve unknown distributions. The problem is to sfely distinguish between these two modes, stnd still nd swing. It is most importnt to minimize the stnd still flse positives, i.e. clling stnd still when the foot is in midir.
3 3.2 Test Sttistic To be ble to differentite between the two modes, test sttistics with known distributions re computed. Three different ones re evluted, one using only the ccelerometer dt, T, one using only the ngulr velocity dt, T ω, nd one using combintion of both ccelerometer nd ngulr velocity dt, T c Accelertion Mgnitude Detector The ccelerometer mgnitude detector test sttistic is computed s Tt = y t 2 σ 2 (4) where T χ 2 (3, λ) during stnd still. It hs noncentrl chi-squre distribution since y t hs nonzero men when the foot is sttionry. Its noncentrlity prmeter λ = g 2 /σ 2 nd 3 is the number of degrees of freedom Angulr Rte Mgnitude Detector The ngulr velocity test sttistic is T ω t = yω t 2 σ 2 ω (5) where T ω χ 2 (3) during stnd still since y ω hs zero men when the foot is sttionry Combined Accelertion nd Angulr Rte Detector The lst test sttistic combines ccelertion nd ngulr velocity to incorporte more informtion. It is clculted s T c t = y t 2 σ 2 + yω t 2 σ 2 ω (6) where T c χ 2 (6, λ) during stnd still. λ is the sme s in (4) but the number of degrees of freedom hs doubled to Test Sttistic Appernce during Wlking Sequence A plot of the test sttistics of the wlking sequence in Figure 2 cn be seen in Figure 3. The stnd still events occuring round time instnts 55, 66, 77, 87 nd 98 re clerly visble. Figure 4 shows zoom in of the test sttistic with the men of the stnd still distribution mrked with dotted line. This revels some of the problems with using only ccelertion for stnd still detection. The test sttistic T hs movement distribution tht hs significnt overlp of the stnd still distribution, cusing the test sttistic to cross the men of the stnd still distribution during the stride. This is shown round time instnts 53, 65, 63, 7, 75, 825 nd 935. Simply clling stnd still when T is close to the men of the stnd still distribution will therefore cuse lot of flse positives T T w T c Figure 3: Test sttistic from top to bottom; Tt, Tt ω nd Tt c. The foot is sttionry round time instnts 55, 66, 77, 87 nd T 5 5 T w 5 T c Figure 4: Zoom in of the test sttistics with the men of the stnd still distribution mrked with dotted line. The foot is sttionry round time instnts 55, 66, 77, 87 nd 98. T ω hs distribution during movement tht does not hve significnt overlp of the stnd still distribution, mking T ω sfer test sttistic thn T to use for stnd still detection. Still, there re two occsions during the stride where the foot is quite sttionry considering the ngulr velocity; one just fter the foot hs been lifted, in Figure 4 shown round time instnts 6, 7, 85 nd 92, nd one just before set down shown t time instnts 525, 635, 745 nd 96. These cn result in flse positives. The third test sttistic T c combines the strengths of T nd T ω. The bottom plot in Figure 4 shows tht the foot does not pper sttionry during the stride when you look t ccelertion nd ngulr velocity simultneously. This results in robust stnd still detection.
4 4 Test Sttistic Distribution Vlidtion The test sttistics must be vlidted to ensure tht the distribution of the test sttistic under experimentl stnd still is close to the theoreticl stnd still distribution. We lso estimte the distribution of the test sttistic under experimentl movements to illustrte the empiricl probbility density functions of stnd still nd movement tht need to be seprted. 4. Accelertion Mgnitude Detector The distributions of the ccelertion mgnitude test sttistic T is shown in Figure 5. The theoreticl nd the empiricl stnd still distributions hve similr men but slightly different covrinces. One of the resons why the empiricl density hs smller covrince thn the theoreticl one, could be tht we hve been bit too meticulous selecting the stnd still dt. Note lso the significnt overlp of the probbility distributions of stnd still nd movement. Tht mkes it difficult to sfely identify stnd stills by only looking t T. p(t ) Probbility Density Function, T 2 3 p(t stnd still) estimted p(t stnd still) estimted p(t moving) p(t ω ) Probbility Density Function, T ω T ω p(t ω stnd still) estimted p(t ω stnd still) estimted p(t ω moving) Figure 6: Theoreticl stnd still distribution of T ω, empiricl estimte of stnd still distribution of T ω from experimentl dt nd empiricl estimte of movement distribution of T ω from experimentl dt. 4.3 Combined Accelertion nd Angulr Rte Detector The combined test sttistic nturlly hs distributions tht look like combintions of the distributions of T nd T ω. The empiricl stnd still distribution hs similr men but slighly smller covrince compred to the theoreticl distribution. The empiricl movement distribution does not overlp the stnd still distributions s much s for T, enbling sfer stnd still detection T Probbility Density Function, T c p(t c stnd still) estimted p(t c stnd still) estimted p(t c moving) 2 Figure 5: Theoreticl stnd still distribution of T, empiricl estimte of stnd still distribution of T from experimentl dt nd empiricl estimte of movement distribution of T from experimentl dt. p(t c ) Angulr Rte Mgnitude Detector The distributions of T ω is shown in Figure 6. Clerly, the theoreticl stnd still distribution is very similr to the empiricl one, estimted by experimentl dt. Also note the lrge seprtion in mgnitude of the empiricl stnd still nd moving distributions. This enbles more robust stnd still detection thn the distributions of T Figure 7: Theoreticl stnd still distribution of T c, empiricl estimte of stnd still distribution of T c from experimentl dt nd empiricl estimte of movement distribution of T c from experimentl dt. T c
5 5 Hidden Mrkov Model To determine the probbility of stnd still, Hidden Mrkov Model (HMM) is used. It determines the probbility of ech mode using the test sttistic, the probbility density functions of the modes nd the mode trnsition probbility mtrix. There re two modes; mode when the foot is t stnd still nd mode 2 when the foot is moving. The mode trnsition probbility mtrix sttes the probbility of mode switch which induces some dynmics into the probbility estimtion. A lower mode trnsition probbility requires mesurement with higher likelihood for the other mode to induce switch. The mode trnsition probbility mtrix is [ ].95.5 Π = (7).5.95 which sttes tht the probility of going from stnd still to moving or vice vers, is 5%. During norml wlking your right foot tkes bout one step per second which results in roughly 2 mode trnsitions every mesurements. The trnsition probbilities were chosen slightly higher to incorporte lso fster movements. The mode probbilities t time t re clculted using the recursion Hence we hve µ i t = µ i t = P (r t = i y t ) p(y t r t = i)p (r t = i y t ) N r = p(t t r t = i) Π ji µ j t. (8) j= p(t t r t = i) N r j= Π jiµ j t Nr l= p(t t r t = l) N r j= Π jlµ j. (9) t The probbility density function of movement used in the HMM is n pproximtion tht is set to resemble the empiricl movement density functions in Figures 5, 6 nd 7. The HMM frmework thus gives the probbility of movement nd stnd still for ech time instnt. This frmework cn be extended to other modes like running nd crwling, simply by extending the mode trnsition probbility mtrix by incorporting these new modes nd estimting the probbility densities for these movements too. 6 Experimentl Results The mode probbilities provided by the HMM of the dt sequence in Figure 2 is shown in Figure 8. All three test sttistics hve been used to illustrte the difference in stnd still detection performnce. The ccelertion bsed test sttistic T suffers from flse positives round some of the troublesome time instnts mentioned in Section 3.3; 65, 7, 75, 825 nd Mode probbility using T Mode probbility using T ω.99 Mode probbility using T c Figure 8: Mode probbilities for the different test sttistics, evluted on the dt set in Figure 2. The foot is sttionry round time instnts 55, 66, 77, 87 nd The frmework does not cll it stnd still fter the first troublesome mesurement, but fter couple of mesurements close to the stnd still men the HMM ssumes the foot is t rest. The ngulr velocity bsed test sttistic T ω gives distict detection of every foot stnce. The sttionry moment is rther short but is often followed by shorter second sttionry moment. Figure 4 shows tht this is becuse there is commonly slight ngulr movement hlfwy through the deemed sttionry prt. This second sttionry moment provides no new informtion nd only the first detection is necessry to perform ZUPT. No flse positives occur during the stride phse of the step. The combined test sttistic T c provides very sfe stnd still detections. A long intervl when the foot is t rest is deemed sttionry nd there re no flse positives. A second dt set is constituted of running phse followed by wlking phse, see Figure 9. The subject is running up until round time instnt. The foot is sttionry round time instnts 725, 8, 9, 5 nd 3, the lst two re during wlking. The mode probbilities provided by the HMM of this sequence is shown in Figure. The sme movement distribution ws used during this whole experiment. T does not provide ny relible stnd still detections. The foot stnces re detected, but lot of flse positives re lso present. This is not surprising considering the ccelerometer dt in Figure 9. The gyro bsed T ω does not result in ny stnd still detections t ll during the running phse. This is bit surprising since the gyro dt looks pretty comprehendble nd is probbly becuse the IMU ws fstened on the side of the foot where only very short periods of low ngulr
6 2 2 3 Accelerometer dt Gyro dt Mode probbility using T Mode probbility using T ω Mode probbility using T c Figure 9: Exmple of ccelerometer nd gyro dt during sequence contining running, 65, followed by wlking, 5. The foot is sttionry round time instnts 725, 8, 9, 5 nd 3. The dt hs the sme color encoding s in Figure 2. velocity re experienced. The combined test sttistic T c still provides quite sfe stnd still detections. It picks up ll the stnd still sequences reveled by ccelertion but mnges to disregrd the flse ones using the ngulr velocity mesurements. Here, the combined test sttistic hs shown to provide the most robust stnd still detection. Further wlking experiments revel the stnd still detection performnce shown in Tble. All 74 true sttionry phses were detected, but lso some flse positives. The ccelertion bsed test sttistic hs flse positive between pretty much every step. Most flse positives of the T c sttistic occur during sequences when smll movements re performed like when door is opened. During wlking, T ω gives very few flse positives nd is the sfest stnd still detector. T T ω T c Stnd stills detected Flse positives Tble : True detected sttionry phses nd flse detected sttionry phses. 74 steps were tken. 6. Step Length Estimtion The foot mounted IMU hs coordinte system following the moving foot. In order to estimte the step length, the orienttion of the foot in world coordintes is described by the unit quternion q. This reltes the mesured ccelertions y t nd ngulr velocities y ω t to movements nd heding chnges in the world coordintes. A filter with the sttes p nd v for position nd velocity in world coordintes nd q cn now be used Figure : Mode probbilities for the different test sttistics during the running followed by wlking sequence. The foot is sttionry round time instnts 725, 8, 9, 5 nd 3. to estimte the length of ech step. For thorough description of the dynmicl model, see [7]. A short experiment of 6 steps covering 5. meters ws performed to evlute the step length estimtion performnce. Stnd still ws detected using the gyro bsed test sttistic nd ZUPT ws performed. Figure shows the estimted movement in world coordintes. Totl step length ws estimted s 4.7 meters rendering step length estimtion error of 6%. position, [m] Movement Estimtion Figure : Movement estimtion in world coordintes. 6 short steps were tken with totl length of 5. meters. 7 Conclusions nd Future Work Three test sttistics with known distributions hve been evluted for stnd still detection. The one bsed on ccelerometer dt only, hs been shown to provide x y z
7 plenty of flse detections. This is nturl since there is significnt overlp between the test sttistic pdf during stnd still nd the pdf during movements. The gyro bsed hs been shown to provide excellent stnd still detection cpbilities during wlking while the one combining ccelerometer nd gyro dt hs been shown to provide good stnd still detection during both wlking nd running. In conjunction with Hidden Mrkov Model, the mode probbilities re redily clculted nd cn be used for zero velocity updtes. Future work includes extending the HMM frmework to incorporte more modes nd to merge the stnd still detection with our locliztion frmework. We will lso look into whether the sttionry phses detected using T c re unnecessrily long for ZUPT. Wht we wnt to detect is gyro drift when the gyro is sttionry nd wht we detect is when the combintion of gyro nd ccelerometer is sttionry. It is not necessrily the sme thing. Further reserch is needed to decide when to perform the zero velocity updte bsed on T c. References [] S. Beuregrd. Omnidirectionl pedestrin nvigtion for first responders. In Proc. of the 4th Workshop on Positioning, Nvigtion nd Communiction, WPNC7, Hnnover, Germny, 27. [2] R. Feliz, E. Zlm, nd J. G. Grci-Bermejo. Pedestrin trcking using inertil sensors. Journl of Physicl Agents, 3():35 43, 29. [3] E. Foxlin. Pedestrin trcking with shoe-mounted inertil sensors. IEEE Computer Grphics nd Applictions, 25(6):38 46, 25. [4] S. Godh, G. Lchpelle, nd M. E. Cnnon. Integrted GPS/INS system for pedestrin nvigtion in signl degrded environment. In Proc. of ION GNSS, 26. [5] L. Ojed nd J. Borenstein. Non-GPS nvigtion for security personnel nd first responders. Journl of Nvigtion, 6(3):39 47, 27. [6] I. Skog. Low-cost Nvigtion Systems. PhD thesis, KTH, Stockholm, Sweden, 29. [7] D. Törnqvist. Estimtion nd Detection with Applictions to Nvigtion. PhD thesis, Linköping University, Linköping, Sweden, 28. [8] O. Woodmn nd R. Hrle. Rf-bsed initilistion for inertil pedestrin trcking. In Proceedings of the 7th Interntionl Conference on Pervsive Computing, 29.
Tests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationData Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading
Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method
More informationDriving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d
Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,
More informationThe steps of the hypothesis test
ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationSection 11.5 Estimation of difference of two proportions
ection.5 Estimtion of difference of two proportions As seen in estimtion of difference of two mens for nonnorml popultion bsed on lrge smple sizes, one cn use CLT in the pproximtion of the distribution
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationLecture 21: Order statistics
Lecture : Order sttistics Suppose we hve N mesurements of sclr, x i =, N Tke ll mesurements nd sort them into scending order x x x 3 x N Define the mesured running integrl S N (x) = 0 for x < x = i/n for
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationFig. 1. Open-Loop and Closed-Loop Systems with Plant Variations
ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses
More informationNon-Linear & Logistic Regression
Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationAN020. a a a. cos. cos. cos. Orientations and Rotations. Introduction. Orientations
AN020 Orienttions nd Rottions Introduction The fct tht ccelerometers re sensitive to the grvittionl force on the device llows them to be used to determine the ttitude of the sensor with respect to the
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationUNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction
Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from
More informationA Signal-Level Fusion Model for Image-Based Change Detection in DARPA's Dynamic Database System
SPIE Aerosense 001 Conference on Signl Processing, Sensor Fusion, nd Trget Recognition X, April 16-0, Orlndo FL. (Minor errors in published version corrected.) A Signl-Level Fusion Model for Imge-Bsed
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS
ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS F. Tkeo 1 nd M. Sk 1 Hchinohe Ntionl College of Technology, Hchinohe, Jpn; Tohoku University, Sendi, Jpn Abstrct:
More informationPredict Global Earth Temperature using Linier Regression
Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationChapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses
Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of
More informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More informationReinforcement learning II
CS 1675 Introduction to Mchine Lerning Lecture 26 Reinforcement lerning II Milos Huskrecht milos@cs.pitt.edu 5329 Sennott Squre Reinforcement lerning Bsics: Input x Lerner Output Reinforcement r Critic
More informationTHREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
More informationTerminal Velocity and Raindrop Growth
Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,
More information2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationHybrid Group Acceptance Sampling Plan Based on Size Biased Lomax Model
Mthemtics nd Sttistics 2(3): 137-141, 2014 DOI: 10.13189/ms.2014.020305 http://www.hrpub.org Hybrid Group Acceptnce Smpling Pln Bsed on Size Bised Lomx Model R. Subb Ro 1,*, A. Ng Durgmmb 2, R.R.L. Kntm
More informationThe Properties of Stars
10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product
More informationResearch on Modeling and Compensating Method of Random Drift of MEMS Gyroscope
01 4th Interntionl Conference on Signl Processing Systems (ICSPS 01) IPCSIT vol. 58 (01) (01) IACSIT Press, Singpore DOI: 10.7763/IPCSIT.01.V58.9 Reserch on Modeling nd Compensting Method of Rndom Drift
More informationX Z Y Table 1: Possibles values for Y = XZ. 1, p
ECE 534: Elements of Informtion Theory, Fll 00 Homework 7 Solutions ll by Kenneth Plcio Bus October 4, 00. Problem 7.3. Binry multiplier chnnel () Consider the chnnel Y = XZ, where X nd Z re independent
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationLECTURE NOTE #12 PROF. ALAN YUILLE
LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, K-mens, nd EM Tsk: set of unlbeled dt D = {x 1,..., x n } Decompose into clsses w 1,..., w M where M is unknown. Lern clss models p(x w)) Discovery of
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationSolution for Assignment 1 : Intro to Probability and Statistics, PAC learning
Solution for Assignment 1 : Intro to Probbility nd Sttistics, PAC lerning 10-701/15-781: Mchine Lerning (Fll 004) Due: Sept. 30th 004, Thursdy, Strt of clss Question 1. Bsic Probbility ( 18 pts) 1.1 (
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationMath 116 Final Exam April 26, 2013
Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationProbability Distributions for Gradient Directions in Uncertain 3D Scalar Fields
Technicl Report 7.8. Technische Universität München Probbility Distributions for Grdient Directions in Uncertin 3D Sclr Fields Tobis Pfffelmoser, Mihel Mihi, nd Rüdiger Westermnn Computer Grphics nd Visuliztion
More informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,
More informationTHERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION
XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More informationMath 32B Discussion Session Session 7 Notes August 28, 2018
Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationA Brief Review on Akkar, Sandikkaya and Bommer (ASB13) GMPE
Southwestern U.S. Ground Motion Chrcteriztion Senior Seismic Hzrd Anlysis Committee Level 3 Workshop #2 October 22-24, 2013 A Brief Review on Akkr, Sndikky nd Bommer (ASB13 GMPE Sinn Akkr Deprtment of
More informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the
More informationDistance And Velocity
Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationCredibility Hypothesis Testing of Fuzzy Triangular Distributions
666663 Journl of Uncertin Systems Vol.9, No., pp.6-74, 5 Online t: www.jus.org.uk Credibility Hypothesis Testing of Fuzzy Tringulr Distributions S. Smpth, B. Rmy Received April 3; Revised 4 April 4 Abstrct
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More informationSurface Integrals of Vector Fields
Mth 32B iscussion ession Week 7 Notes Februry 21 nd 23, 2017 In lst week s notes we introduced surfce integrls, integrting sclr-vlued functions over prmetrized surfces. As with our previous integrls, we
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationMonte Carlo method in solving numerical integration and differential equation
Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationPurpose of the experiment
Newton s Lws II PES 6 Advnced Physics Lb I Purpose of the experiment Exmine two cses using Newton s Lws. Sttic ( = 0) Dynmic ( 0) fyi fyi Did you know tht the longest recorded flight of chicken is thirteen
More informationProbabilistic Investigation of Sensitivities of Advanced Test- Analysis Model Correlation Methods
Probbilistic Investigtion of Sensitivities of Advnced Test- Anlysis Model Correltion Methods Liz Bergmn, Mtthew S. Allen, nd Dniel C. Kmmer Dept. of Engineering Physics University of Wisconsin-Mdison Rndll
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationThe Fundamental Theorem of Calculus, Particle Motion, and Average Value
The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt
More informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationModule 2: Rate Law & Stoichiomtery (Chapter 3, Fogler)
CHE 309: Chemicl Rection Engineering Lecture-8 Module 2: Rte Lw & Stoichiomtery (Chpter 3, Fogler) Topics to be covered in tody s lecture Thermodynmics nd Kinetics Rection rtes for reversible rections
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationTutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits.
Tutoril 4 Exercises on Differentil Entropy. Evlute the differentil entropy h(x) f ln f for the following: () The uniform distribution, f(x) b. (b) The exponentil density, f(x) λe λx, x 0. (c) The Lplce
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationStudent Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationMinimum Energy State of Plasmas with an Internal Transport Barrier
Minimum Energy Stte of Plsms with n Internl Trnsport Brrier T. Tmno ), I. Ktnum ), Y. Skmoto ) ) Formerly, Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Plsm Reserch Center, University
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationFlexible Beam. Objectives
Flexile Bem Ojectives The ojective of this l is to lern out the chllenges posed y resonnces in feedck systems. An intuitive understnding will e gined through the mnul control of flexile em resemling lrge
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More information